On right eigenvalues inverse problem of the circulant matrix over quaternion division ring

The circulant matrices have special effects in researching of the control system and other fields. This paper mainly discusses the right eigenvalues inverse problem of the quaternion circulant matrix. We firstly introduce the definition of a quaternion standard element by the characteristics of quaternions similar classes. Next, for any given n quaternions, by using complex representation of a quaternion matrix, it is proved that there exists a quaternion circulant matrix A such that the n quaternions are the right eigenvalues of A. Meanwhile, the expression of general solution for this problem is given.